5. Nonlinear Theory: Guided-Mode Analysis

The self-consistent nonlinear theory of the free-electron laser describes the interaction through the linear regime and includes the saturation of the growth mechanism. Saturation can occur through a variety of mechanisms. For an ideal beam that is both monoenergetic and vanishing pitch-angle spread, saturation occurs by means of electron trapping in the ponderomotive potential. In the thermal regime, saturation occurs by a different process. In this case, the axial energy spread of the beam (which can arise due to either a distribution in the total energy of the beam electrons or pitch-angle spread) gives rise to a broadband emission spectrum. As a result, a quasilinear saturation mechanism is operative in which the beam undergoes turbulent diffusion in momentum space. The growth rate in this regime is proportional to the slope of the distribution function; turbulent diffusion acts to form a plateau in momentum space that flattens out the distribution of the beam. As a result, the axial energy spread of the beam increases, and the instability is quenched when the slope of the distribution falls to zero. However, the saturation efficiency in the thermal regime is greatly reduced relative to that found for a sufficiently cold beam in which saturation occurs through the particle-trapping mechanism. We shall focus attention on the latter case in this chapter. This chapter will describe the development of slowly varying envelope approximation (SVEA) formulations in the steady-state regime, as well as the application of the analyses to the description of the fundamental physics of the nonlinear saturation mechanism.